Prove that if a , b , c &#x2208;<!-- ∈ --> <mrow class="MJX-TeXAtom-ORD">

Step 1
The first part of your proof is enough because you can proceed by equivalence
$\text{aligned points}⟺\text{equal slopes}⟺c^2+bc+b^2=b^2+ab+a^2⟺(a-c)(a+b+c)=0⟺(a+b+c)=0$
Here is an alternate proof:
Using the classical alignment criteria for 3 points $(x_k,y_k)$ for k=1,2,3 which is
$\left|\begin{array}{ccc}{x}_1& x_2& x_3\\ y_1& y_2& y_3\\ 1& 1& 1\end{array}\right|=0$
This means that we have to show that
$\left|\begin{array}{ccc}a& b& c\\ a^3& b^3& c^3\\ 1& 1& 1\end{array}\right|=0⟺a+b+c=0$
But this is very easy because the determinant can be factorized in the following way:
$(a-c)(b-a)(b-c)(a+b+c),$
knowing that a,b,c are all different.
In fact, I just realized that I had already answered a similar question here... with two proofs, this one and another one based on a third degree equation with no term in $x^2$